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Mathematics (Pomona)

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Complex symmetric operator, Takagi factorization, inner function, Aleksandrov-Clark operator, Clark operator, Aleksandrov measure, compressed shift, Jordan operator, $J$-selfadjoint operator, Sturm-Liouville problem


A bounded linear operator T on a complex Hilbert space H is called complex symmetric if T = CT*C, where C is a conjugation (an isometric, antilinear involution of H). We prove that T = CJ|T|, where J is an auxiliary conjugation commuting with |T| = √{T*T). We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition T = CJ|T| also extends to the class of unbounded C-self adjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms of the resolvents of certain unbounded operators.


First published in Transactions of the American Mathematical Society in Volume 359, Number 8, August 2007, published by the American Mathematical Society.

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© 2007 American Mathematical Society

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